Weather forecasts are of high value to a wide range of activities in agriculture, energy, transportation, defense and supply chain management. See K. Rollins and J. Shaykewich, “Using willingness-to-pay to assess the economic value of weather forecasts for multiple commercial sectors,” Meteorol. Appl. Vol. 10, pp. 31-38 (2003); Y. Zhu, Z. Toth, R. Wobus, D. Richardson, and K, Mylne, “The Economic Value of Ensemble-Based Weather Forecasts,” BAMS, January 2003, pp. 73-83. Of particular value are weather forecasts tailored for the particular weather related decisions that individuals, companies and organizations must make.
Such tailored weather forecasts are often provided by private weather companies such as those listed on the government web site “www(dot)weather(dot)gov/im/more.htm”.
The starting point for these weather forecasts is a computer simulation or Numerical Weather Prediction (NWP) in which approximations to the equations governing the evolution of the atmosphere propagate a recently estimated weather state forward in time. In general, substantial improvements to forecasts made in this way can be achieved by comparing histories of forecasts with corresponding histories of verifying observations. By mathematically modeling the differences between the NWP and observations, it has been shown that one can produce a forecast that is significantly better than that from NWP.
Glahn and Lowry describe how the theory of multivariate linear regression can be used to correct systematically predictable aspects of weather forecast models such as the NWP, and refer to their weather forecast error corrector as Model Output Statistics (MOS). See H. R. Glahn and D. A. Lowry, “The Use of Model Output Statistics (MOS) in Objective Weather Forecasting,” J. Appl. Meteor., Vol. 11, Issue 8, pp. 1203-1211 (1972). The MOS forecast error corrector is estimated from a historical record of forecasts together with corresponding verifying observations. Wilson and Vallée (2002) suggest that the historical record needs to contain at least two years worth of daily weather forecasts. L. J. Wilson and M. Vallée, “The Canadian Updateable Model Output Statistics (UMOS) System: Design and Development Tests,” Wea. Forecasting, Vol. 17, Issue 2, pp. 206-222 (2002).
MOS methods assume that on the kth day or event, the ith model forecast variable fik is related to the corresponding verifying analysis or observation variable yik via the stochastic relationfik=aiyik+bi+εik  (1)where the true regression coefficients ai and bi of the model forecast variable fik can be considered to be the “slope” parameter and the “intercept” parameter, respectively, of a plot of the relation between f and y, and εik is a random number associated with the kth forecast that is statistically independent of yik.
With a finite sample of k=1, M realizations, using MOS methods one can obtain estimates aiMOS and biMOS of the regression coefficients ai and bi using the relations
                                          a            i            MOS                    =                                    [                                                ∑                                      k                    =                    1                                    M                                ⁢                                                                  ⁢                                                      (                                                                  f                        i                        k                                            -                                                                        f                          _                                                i                                                              )                                    ⁢                                      (                                                                  y                        i                        k                                            -                                                                        y                          _                                                i                                                              )                                                              ]                                                      ∑                                  k                  =                  1                                M                            ⁢                                                          ⁢                                                (                                                            y                      i                      k                                        -                                                                  y                        _                                            i                                                        )                                ⁢                                  (                                                            y                      i                      k                                        -                                                                  y                        _                                            i                                                        )                                                                    ,                                  ⁢                              b            i            MOS                    =                                                    f                _                            i                        -                                          a                i                MOS                            ⁢                                                y                  _                                i                                                    ,                                  ⁢                              where            ⁢                                                  ⁢                                          f                _                            i                                =                                    1              M                        ⁢                                          ∑                                  k                  =                  1                                M                            ⁢                              f                i                k                                                    ,                                            y              _                        i                    =                                    1              M                        ⁢                                          ∑                                  k                  =                  1                                M                            ⁢                              y                i                k                                                                        (        2        )            where fik is the ith model forecast variable and yik is the verifying analysis/observation variable in Equation (1) above and fi and yi are the average values over M realizations.
If two years of historical data are available for the forecast model of interest, MOS can be relied on to provide significant forecast improvements. However, in the continuing effort to improve weather forecasting models, significant changes are frequently made to the forecasting model. The MOS equations developed for an old model may be entirely inappropriate for the changed new model. Consequently, after every model change, new MOS equations need to be developed. For example, typical model changes have included increases in model resolution, changes in the representation of sub sub-grid scale physics, and changes in the forcings associated with radiation.
If sufficient manpower and computer resources were available, a historical record of the performance of the new system required for new MOS equations could be generated fairly quickly by running the new system on historical data. However, such resources are generally unavailable to the responsible parties. Instead, a historical record of two years is usually obtained by archiving forecasts and the corresponding verifying analyses and forecasts in real time. Using such a method, a two-year historical record would require two years' time to compile.
This state of affairs led Wilson and Vallée (2002), supra, to propose an Updateable Model Output Statistics (UMOS) system that blended regression equations based on earlier versions of the model with equations based on more recent versions of the model. UMOS implicitly assumes that significant model changes are typically separated by more than two years. However, this assumption is not satisfied at many forecasting centers. For example, the limited area weather forecasting models of the U.S. Navy are typically deployed in a specific theatre for a period between two weeks and two years. In principle, new MOS equations would need to be developed for each unique theatre in which the forecasting model is deployed. Hence, the Navy's deployment periods are too short for the estimated regression coefficients to stabilize, and any attempt to estimate the regression coefficients using MOS or UMOS with such a small data set will result in very noisy regression coefficients.
One alternative would be to assume that regression coefficients were identical in various sub-regions of the model. This was assumed by, for example, M. J. Schmeits, K. J. Kok, and D. H. P. Vogelezang, “Probabilistic Forecasting of (Severe) Thunderstorms in the Netherlands Using Model Output Statistics.” Wea. Forecasting, Vol. 20, Issue 2, pp. 134-148 (2005). Although this approach will give stable coefficients for small data training sets it does not converge to the (superior) MOS coefficients in the limit of infinite training data. In addition, the MOS equations are likely to have erroneous jumps between sub-regions.
Thus, there is a need for a method for producing less noisy MOS regression coefficients that can be used with small data sets.